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Page 106
... simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product Zef of ƒ with the characteristic function ... functions are M ( S , E , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
... simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product Zef of ƒ with the characteristic function ... functions are M ( S , E , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
Page 108
... simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n S_h ( s ) μ ( ds ) = S_ f ( s ) u ( ds ) = · S2 ...
... simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n S_h ( s ) μ ( ds ) = S_ f ( s ) u ( ds ) = · S2 ...
Page 322
... function ƒ defined on S is μ - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
... function ƒ defined on S is μ - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ